Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











HypergeometricU






Mathematica Notation

Traditional Notation









Hypergeometric Functions > HypergeometricU[a,b,z] > Specific values > For fixed z > For fixed z and a=-1/2





http://functions.wolfram.com/07.33.03.0287.01









  


  










Input Form





HypergeometricU[-(1/2), -(11/2), -z] == (1/(7680 Sqrt[z])) ((Sqrt[z] (2 E^z Sqrt[-z] (10395 + 8 z (315 + 2 z (42 + z (8 + z)))) + Sqrt[Pi] (10395 + 2 z (4725 + 4 z (525 + 2 z (75 + z (15 + 2 z)))))) - Sqrt[Pi] Sqrt[-z] (10395 + 2 z (4725 + 4 z (525 + 2 z (75 + z (15 + 2 z))))) Erfi[Sqrt[z]])/E^z)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["7680", " ", SqrtBox["z"]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", "z"], " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List["10395", "+", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List["315", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["42", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["8", "+", "z"]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["10395", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["4725", "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["525", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["75", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["15", "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List["10395", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["4725", "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["525", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["75", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["15", "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]]]]]], ")"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> U </mi> <annotation encoding='Mathematica'> TagBox[&quot;U&quot;, HypergeometricU] </annotation> </semantics> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 11 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mn> 7680 </mn> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 8 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 8 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 42 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 315 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 10395 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 75 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 525 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4725 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 10395 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 15 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 75 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 525 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 4725 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mn> 10395 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> erfi </mi> <mo> &#8289; </mo> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricU </ci> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 11 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 7680 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 8 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <cn type='integer'> 42 </cn> </apply> </apply> <cn type='integer'> 315 </cn> </apply> </apply> <cn type='integer'> 10395 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 15 </cn> </apply> </apply> <cn type='integer'> 75 </cn> </apply> </apply> <cn type='integer'> 525 </cn> </apply> </apply> <cn type='integer'> 4725 </cn> </apply> </apply> <cn type='integer'> 10395 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> 15 </cn> </apply> </apply> <cn type='integer'> 75 </cn> </apply> </apply> <cn type='integer'> 525 </cn> </apply> </apply> <cn type='integer'> 4725 </cn> </apply> </apply> <cn type='integer'> 10395 </cn> </apply> <apply> <ci> Erfi </ci> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HypergeometricU", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], ",", RowBox[List["-", FractionBox["11", "2"]]], ",", RowBox[List["-", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", "z"], " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List["10395", "+", RowBox[List["8", " ", "z", " ", RowBox[List["(", RowBox[List["315", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["42", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["8", "+", "z"]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "+", RowBox[List[SqrtBox["\[Pi]"], " ", RowBox[List["(", RowBox[List["10395", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["4725", "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["525", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["75", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["15", "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]], "-", RowBox[List[SqrtBox["\[Pi]"], " ", SqrtBox[RowBox[List["-", "z"]]], " ", RowBox[List["(", RowBox[List["10395", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["4725", "+", RowBox[List["4", " ", "z", " ", RowBox[List["(", RowBox[List["525", "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List["75", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["15", "+", RowBox[List["2", " ", "z"]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]], " ", RowBox[List["Erfi", "[", SqrtBox["z"], "]"]]]]]], ")"]]]], RowBox[List["7680", " ", SqrtBox["z"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02